Theorem seqeq1 10713

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq1	|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Proof of Theorem seqeq1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 |- ( M = N -> M = N )
2		fveq2 5639	. . . . . 6 |- ( M = N -> ( F ` M ) = ( F ` N ) )
3	1, 2	opeq12d 3870	. . . . 5 |- ( M = N -> <. M , ( F ` M ) >. = <. N , ( F ` N ) >. )
4		freceq2 6559	. . . . 5 |- ( <. M , ( F ` M ) >. = <. N , ( F ` N ) >. -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
5	3, 4	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
6		fveq2 5639	. . . . . 6 |- ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
7		eqid 2231	. . . . . 6 |- _V = _V
8		mpoeq12 6081	. . . . . 6 |- ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ _V = _V ) -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
9	6, 7, 8	sylancl 413	. . . . 5 |- ( M = N -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
10		freceq1 6558	. . . . 5 |- ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
11	9, 10	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
12	5, 11	eqtrd 2264	. . 3 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
13	12	rneqd 4961	. 2 |- ( M = N -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
14		df-seqfrec 10711	. 2 |- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
15		df-seqfrec 10711	. 2 |- seq N ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. )
16	13, 14, 15	3eqtr4g 2289	1 |- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Syntax hints:   -> wi 4   = wceq 1397   _Vcvv 2802   <.cop 3672   ran crn 4726   ` cfv 5326  (class class class)co 6018   e. cmpo 6020  freccfrec 6556   1c1 8033   + caddc 8035   ZZ>=cuz 9755   seqcseq 10710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fv 5334  df-oprab 6022  df-mpo 6023  df-recs 6471  df-frec 6557  df-seqfrec 10711

This theorem is referenced by:  seqeq1d  10716  seq3f1olemqsum  10776  seqf1oglem2  10783  seq3id  10788  seq3z  10791  iserex  11917  summodclem2  11961  summodc  11962  zsumdc  11963  isumsplit  12070  ntrivcvgap  12127  ntrivcvgap0  12128  prodmodclem2  12156  prodmodc  12157  zproddc  12158  fprodntrivap  12163  ege2le3  12250  gsumfzval  13492  gsumval2  13498